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www.jeremykun.com | ||
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www.math3ma.com
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| | | | | [AI summary] The article explains the hierarchy of integral domains in abstract algebra, detailing the relationships and proofs between fields, Euclidean domains, principal ideal domains, and unique factorization domains. | |
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jaydaigle.net
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| | | | | We continue our exploration of what numbers are, and where mathematicians keep finding weird ones. In the first three parts we extended the natural numbers in two ways: algebraically and analytically. Those approaches gave overlapping but distinct sets of numbers. This week we combine them to get the complex numbers, and see some hints of why the complex numbers are so useful-and so frustrating. | |
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mattbaker.blog
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| | | | | In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I'd like to explain how the uniqueness portion of that theorem is actually a special case of a more general... | |
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www.ethanepperly.com
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| | | [AI summary] The user has provided a comprehensive set of mathematical derivations and inequalities related to probability and statistics, focusing on bounding the probability that the sample mean deviates from its expected value. The key points include: (1) Using the Chernoff bound to derive a tail inequality for bounded random variables, (2) Applying the Hoeffding's inequality to show that the probability of the sample mean deviating by more than t is exponentially small in n, (3) Deriving the final result that the probability is bounded by 2exp(-2nt^2/(b-a)^2), and (4) Mentioning the connection to the Hoeffding inequality. The user also included some unrelated text about an ORCID iD and an email profile, which seems to be an error or unrelated information... | ||
